Granular Flows 1
Lecture 6: Flow regimes fluid-like Quasi-static granular flows have plasticity laws, gaseous granular flows have kinetic theory -- how to model fluid-like flows? Intermediate, dense regime: Enduring contacts between particles Existence of a force network Content: Local rheology: friction & dilatancy laws Hydrodynamical approach: shallow-water equations 2
Richtmyer-Meshkov instability (1) One of the classical fluid dynamic instabilities: Impulsive acceleration of a density-differentiated interface: Shock wave creates small amplitude perturbations Nonlinear regime of bubbles or spikes chaotic regime where fluids mix Granular equivalent dropping a tray with sand: initial profile: dips upon impact: reversing Impulse strength: weak: avalanche strong: Richtmyer-Meshkov From: personal communication with Dr. Stuart Dalziel 3
h (mm) h/h Richtmyer-Meshkov instability (2) 20.0 0.4 10.0 0.2 0.0 0.0 10.0 0.2 20.0 0.4 0.0 50.0 100.0 150.0 200.0 x (mm) From: personal communication with Dr. Stuart Dalziel 4
Dense granular flows (1) Let s start with experiments & simulations: GDR Midi (Groupement de Recherché Milieux Divisés) Confined by walls: a) Plane shear b) Couette cell c) Silo Free surface flows: d) Inclined plane e) Pile f) Rotating drum From: GDR Midi, on dense granular flows, Eur Phys J E Soft Matter, 2004 5
Different geometries: Plane shear Two parallel rough walls, one moves: Relation inertia & confining stress:,with: Velocity profile: small I: linear, uniform shear rate dense flow, is uniform large I: slip velocity, S-shaped dilute flow, in centre From: GDR Midi, Eur Phys J E Soft Matter, 2004 6
Different geometries: Couette cell Taylor-Couette cell = annular shear cell: Two rough coaxial cylinders moving inner cylinder, stationary outer cylinder thin (5 10d) shear bands localized at wall hydrostatic pressure, not Janssen effect! Velocity profile does not depend on shear rate: Decays faster than exponential, rather Gaussian except close to wall From: GDR Midi, Eur Phys J E Soft Matter, 2004 7
Different geometries: Silo Cylinder (3D) or two parallel walls (2D): Flow rate controlled by: Outlet aperture dimensions Velocity moving slider If column is high enough: Jansen effect Velocity profile: Centre: plug region (constant velocity) Walls: sheared region (varied velocity) Rescaled velocity profile is independent of flow rate (quasi-steady) From: GDR Midi, Eur Phys J E Soft Matter, 2004 8
Different geometries: Inclined plane (1) Avalanches on an inclined plane: Steady uniform flows: constant V & h Non-steady flow if or h acceleration of flow No flow if h = h stop by decreasing h or h stop () curve: resistance is higher closer to surface From: Pouliquen et al., Physics of Fluids, 1999 9
Focus: Inclined plane (2) Observations on velocity: Continuous transition between inclined plane & surface flow? Thick layers (h >> h stop ): Bagnold velocity profile: accurate in core layer not accurate close to base or free surface (where I is not constant) Thin layers (h ~ h stop ): linear velocity profile: V ~ h Empirical flow rule for depth-averaged velocity <V>: From: Jop et al., 2005, Pouliquen et al., 2006 10
Different geometries: Pile Hele-Shaw cell: between glass plates Avalanches self-adjust: depend on injection flow rate Q high Q: stationary regime (after transient) low Q: intermittent flow Intermittent flow: successive transient avalanches surface slope oscillates between start & stop Localized free-surface flow: linear velocity profile near surface (dotted line) exponentially creeping tail (dashed line) From: Jop et al., 2005, GDR Midi, Eur Phys J E Soft Matter, 2004 11
Different geometries: Rotating drum Rotating drum: Higher flow (= rotating) rates: pronounced lens S-shape free surface maximum layer thickness increases Only homogeneous flow at centre: layer thickness & local slope vary slowly general scaling thickness: Localized free-surface flow: Linear velocity profile near surface dense inertial flow Exponentially creeping tail quasi-static confined flow From: GDR Midi, Eur Phys J E Soft Matter, 2004 12
Local rheology (1) Rheology from dimensional arguments & simulations: Shear stress proportional to pressure: Volume fraction: Inertial number defines flow regime: Microscopic (inertial) time scale: Macroscopic (deformation) time scale: Transition regimes for increasing I: quasi-static dense inertial collisional regime Kinetic regime: Friction not relevant Quasi-static regime: Grain-inertia not relevant From: da Cruz et al., PRE, 2005 & From: Jop et al., JFM, 2005 13
Local rheology (2) Friction & dilatancy laws from empirical evidence: Correct for 2D configurations: plane shear & inclined plane Friction law: Volume fraction: I is rate-dependent in intermediate regime flow law Dissipation dominates sliding: I From: Pouliquen et al., J. of Stat. Mech., 2006 & da Cruz et al., PRE, 2005 14
Local rheology (3) Bagnold s experiments and scaling: For all shear rates and regimes, for perfectly hard grains Normal stress: Shear stress: Ratio: Rewriting friction & volume fraction in terms of f 1 & f 2 : Friction: Volume fraction: f 1 and f 2 diverge quickly near maximum packing fraction friction () and dilatancy () laws are decoupled From: Lois et al., PRE, 2005, Forterre & Pouliquen, Annu. Rev. Fluid Mech., 2008 15
Constitutive law for granular liquids? (1) Constitutive relations? Valid for other geometries? Simulations & experiments: a) Plane-shear b) Rotating drum c) Inclined planes d) Annular shear cell Yes, collapse! Relevant parameter: I a) Theoretical fit (red) & kinetic theory (blue) All dense granular flows: local friction and dilatation laws From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 16
Constitutive law for granular liquids? (2) Rheology for all geometries (not only plane shear): Visco-plastic (Bingham) model (Jop et al., 2006) Flow threshold viscosity instead of yield stress Shear rate dependence viscous behavior Analogy to Bingham fluids Nonlinear elasto-plastic model (Kamrin, 2010), includes: Granular elasticity (Jiang & Liu, 2003) for stagnant zones Rate-sensitive fluid-like flow (Jop et al., 2006) for flowing regions 17
Visco-plastic model (1) 3D geometries -- shear from different directions: Non-Newtonian incompressible fluid: assume volume fraction is constant in dense regions co-linearity between shear stress and shear rate Form of a visco-plastic law: isotropic pressure P shear stress:, with viscosity: second invariant of shear rate tensor Flow threshold (Drucker-Prager criterion): second invariant of stress tensor goes to zero viscosity diverges (difficult in some simulations!) Predicts correctly: critical angle & constant volume fraction Bagnold velocity profile From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 18
Visco-plastic model (2) Limitations on using a visco-plastic approach: Lack of link with microscopic grain properties: shape of friction law is measured, not derived Shear bands (quasi-static regions) are incorrectly described: modifying plasticity models in shear-rate independent regime explicitly describing nonlocal effects (e.g. jamming) Flow threshold: Coulomb criterion, does not capture hysteresis and finite size effects Transition to kinetic regime: gaseous regime is not captured in visco-plastic approach kinetic theory does not capture correct behavior in dense regime Theoretical fit (red) & kinetic theory (blue) From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 19
Shallow water equations (1) Alternative constitutive relation for thin flows: Interfacial law between bottom and granular layer dynamics of flowing layer without knowing details internal structure Depth-averaged or Saint-Venant equations: assuming incompressible flow variations are on a scale larger than flow thickness Mass conservation: Momentum conservation: with basal friction coefficient b, velocity coefficient and stress ratio K From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 20
Shallow water equations (2) Limitations on using shallow water equations: Coulomb-type basal friction may not be sufficient rough inclines steady uniform flow for different inclination angles solid friction is not constant, complicated basal friction laws necessary Second-order effects are not captured: longitudinal and lateral momentum diffusion are not included necessary to control instabilities and lateral stresses Additional equation necessary for erodible layers: exchange of mass and momentum between solid-liquid interface From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 21
Shallow water equations (3) Debris avalanche: Montserrat, December 1997 Failure of south flank of Soufriere Hills volcano numerical simulations: gravitational flow of a homogeneous continuum Coulomb-type basal friction with a dynamic friction coefficient From: Heinrich et al., GRL, 2001 22
Fun examples: different flow regimes (1) Shaking in vertical direction: Displacement: Acceleration: dilation & microscopic movement Processes: Vibration-induced mixing & size segregation Brazil-nut problem more in later lecture Vibration-induced convection and heaping Convection rolls, granular Leidenfrost Surface phenomena: travelling or standing waves Faraday instabilities, Richtmyer-Meshkov instability From: Jaeger, Nagel & Behringer, Reviews of Modern Physics, 1996 23
Fun examples: different flow regimes (2) Vibration-induced vertical shaking Shaking acceleration: Shaking strength: Number of bead layers: F Different regimes: Bouncing bed Undulations (standing waves) Granular Leidenfrost effect Convection rolls Granular gas From: Eshuis et al., Physics of Fluids, 2007 24
Fun examples: different flow regimes (3) Faraday heaping -- surface phenomenon Ambient air plays critical role Stable shape of heaps: internal avalanches horizontal pressure gradients stability inclined surfaces Pressure-gradient mechanism: convection rolls isobars parallel to surface Coarsening process: small initial surface deflections transient state of several heaps merging due to reduced inward drag force From: van Gerner et al., Physical Review E, 2007 25
Fun examples: different flow regimes (4) Quicksand: sand, clay & salt water Sensitive to small stress variations: low stress: viscosity slowly changes high stress: viscosity drops magnitudes liquid behavior reduces friction first liquefies, then collapses Sinking mechanism: trapping: liquefaction and sedimentation water and sand separates in fractions apparent viscosity increases untrapping: add water to liquefy compacted sand alternatively: low density humans prevents complete submersion! From: Bonn et al., Nature, 2005 26
Fun examples: different flow regimes (5) Sandcastles: perfect composition? Stiffness of sand: dry: hardly supports own weight wet: forms liquid bridges (ideal: 1%) too wet: bridges form large liquid pockets Stability of wet sand columns: elastic buckling under own weight critical height: column of R = 20 cm h crit = 2.5 m increase height: compaction or density use hydrophobic sand: water air From: Bonn et al., Nature, 2005, Moller & Bonn, EPL, 2007 27